# Counterexample where X is not in the Lebesgue linear space.

mehr1methanol
Example where X is not in the Lebesgue linear space.

## Homework Statement

I'm trying to find an example where $\lim_{n \to +\infty} P(|X|>n) = 0$ but $X \notin L$ where $L$ is the Lebesgue linear space.

Relevant equations:

$X$ is a random variabel, $P$ is probability. $I$ is indicator function.

The attempt at a solution

$∫|X|I(|X|>n)dp + ∫|X|I(|X|≤n)dp = ∫|X|dp$

$∫nI(|X|>n)dp + ∫|X|I(|X|)dp ≤ ∫|X|dp$

Suppose $∫I(|X|>n)dp = 1/(n ln n)$
Clearly the hypothesis is satisfied because $\lim_{n \to +\infty} P(|X|>n) = \lim_{n \to +\infty} ∫I(|X|>n)dp = \lim_{n \to +\infty} 1/( ln n) = 0$
But I'm not sure how to conclude $∫|X|dp = ∞$

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